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\begin{document}

\title{Stat 992: Lecture 09\\
Simulating Gaussian fields.}
\author{Moo K. Chung \tt{mchung@stat.wisc.edu}}
\maketitle \thispagestyle{empty}

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\begin{enumerate}

\item {\em White noise.} White noise is defined as a random field
whose covariance function is proportional to the Dirac-delta
function $\delta$, i.e. $R(x,y) \propto \delta(x-y)$. For
instance, me may take $R(x,y) = \lim_{\sigma \to 0} K_{\sigma}(\|x
- y\|)$, the limit of the usual isotropic Gaussian kernel. From a
standard mathematical sense, white noise does not exists and it
can be characterized via generalized functions.

One example of white noise is the generalized derivative of
Brownian motion (Wiener process) called white Gaussian noise. 1D
Wiener process $B(x), x \in \mathbb{R}^+$ is zero mean Gaussian
field with covariance function $R_B(x,y) = \min(x,y)$. Based on
this definition, we can show ${\bf Var} B(x) = x$ by taking $x=y$
in the covariance function and $B(0)=0$ by letting $x=0$ in the
variance. Increments of Wiener processes in nonoverlapping
intervals are i.i.d. Gaussian. Further the paths of Wiener process
is continues in the Kolmogorov sense while it is not
differentiable. For a different but identical canonical
construction of Brownian motion, see Bernt Oksendal's Stochastic
Differential Equations (5th edition, 2000). Higher dimensional
Brownian motion can be generalized by taking each component of
vector fields to be i.i.d. Brownian motion.

{\bf Problem 18.} In MATLAB, simulate 1D and 2D version of
Brownian motion based on the above properties and check ${\bf Var}
B(x) = x$ by simulating 10000 Brownian motions. What is the
corresponding 2D version ?


\begin{figure}
\centering
\renewcommand{\baselinestretch}{1}
\includegraphics[scale=0.24]{ima09-1.eps}
\includegraphics[scale=0.24]{ima09-2.eps}
\includegraphics[scale=0.24]{ima09-3.eps}
\includegraphics[scale=0.24]{ima09-4.eps}
\caption{Random fields simulation via iterated Gaussian kernel
smoothing with $\sigma=0.4$. $N(0,0.4^2)$. White noise, 1, 4 and 9
iterations in sequence.\label{fig:lec9-1}}
\end{figure}

\item {\em Gaussian white noise.} Although the path of Wiener
process is not differentiable, we can define the generalized
derivative via integration by parts with a smooth function $f$
called a test function in the following way
$$f(x)B(x) = \int_0^x f(y)\frac{dB(y)}{dy} \;dy + \int_0^x \frac{f(y)}{dy}B(y)\;dy.$$
Taking the expectation on both sides we have
$$ \int_0^x f(y) \mathbb{E}\frac{dB(y)}{dy} \;dy =0.$$
It should be true for all smooth $f$ so $\mathbb{E}
\frac{dB(y)}{dy} =0$. Further it can be shown that the covariance
function of process $dB(y)/dy \propto \delta(x-y)$.

{\bf Problem 19.} Prove this fact and hence showing the
generalized derivative of Brownian motion is a white noise. It is
usually called a Gaussian white noise.

The Gaussian white noise can be used to construct smooth Gaussian
random fields of the form
$$X(x) = K * W(x) = K * \frac{dB(x)}{dx}.$$
Since Brownian motion is zero mean Gaussian process, $X(x)$ is
obviously zero mean field with the covariance function \bqn
R_X(x,y) &=& \mathbb{E} [K * W(x)K * W(y)]\\
&\propto&\int K(x-z)K(y-z) \; dz.\eqn The case $K=K_{\sigma}$
isotropic Gaussian kernel type has been investigated by D.O.
Siegmund and K.J. Worsley with respect to optimal filtering in
scale space theory (Annals of Statistics, 23:608-639, 1995).

\item {\em Numerical Implementation.} In numerical implementation,
we use the discrete white Gaussian noise which is simply a
Gaussian random variable. Let $w$ be a discrete version of white
Gaussian noise given by
$$w(x) = \sum_{i=1}^m Z_i \delta(x-x_i)$$
where i.i.d. $Z_i \sim N(0,\sigma_w^2)$. Note that
$$K* w(x) = \sum_{i=1}^m Z_iK(x-x_i).$$
Trivially the set of random variables $K* w(y_j)$ forms a
multivariate normal at arbitrary points $y_1,\cdots, y_l$. Hence
the field $K * w(x)$ is a Gaussian field. The covariance function
is given by

\bqn R(x,y) &=& \sum_{i,j=1}^m \mathbb{E}(Z_iZ_j) K(x-x_i)
K(y-x_j) \\
&=&\sum_{i=1}^m \sigma_w^2K(x-x_i) K(y-x_i).
 \eqn
 As usual we may take $K$ to be a
Gaussian kernel. Let us simulate some Gaussian fields. Following
the simulation example in lecture 4, the unknown signal is assumed
to be $\mu(x,y) = \cos (10x) + \sin(8y), (x,y) \in [0,1]^2$ and
white noise error $w \sim N(0,0.4^2)$ which gives the left figure
in Figure 1.


\begin{verbatim}
w=normrnd(0,0.4,101,101);
smooth_w=w;
for i=1:10
  smooth_w=conv2(smooth_w,K,'same');
  figure;imagesc(smooth_w);colorbar;
end;
\end{verbatim}


\end{enumerate}

\end{document}


 \item {\em Fourier transform in $\mathbb{R}^n$.} Fourier
transform of $f$ is an integral transform
$$\mathcal{F}(f)(w) = \frac{1}{(2\pi)^{n/2}}\int e^{-iw\cdot t}f(t) \; dt$$
The inversion formula is
$$\mathcal{F}^{-1}(f)(t) = \frac{1}{(2\pi)^{n/2}}\int e^{iw\cdot t}f(w) \; dw$$

We will apply the Fourier transform to